Incompressible Image Registration Using Divergence-Conforming B-Splines

Abstract

Anatomically plausible image registration often requires volumetric preservation. Previous approaches to incompressible image registration have exploited relaxed constraints, ad hoc optimisation methods or practically intractable computational schemes. Divergence-free velocity fields have been used to achieve incompressibility in the continuous domain, although, after discretisation, no guarantees have been provided. In this paper, we introduce stationary velocity fields (SVFs) parameterised by divergence-conforming B-splines in the context of image registration. We demonstrate that sparse linear constraints on the parameters of such divergence-conforming B-Splines SVFs lead to being exactly divergence-free at any point of the continuous spatial domain. In contrast to previous approaches, our framework can easily take advantage of modern solvers for constrained optimisation, symmetric registration approaches, arbitrary image similarity and additional regularisation terms. We study the numerical incompressibility error for the transformation in the case of an Euler integration, which gives theoretical insights on the improved accuracy error over previous methods. We evaluate the proposed framework using synthetically deformed multimodal brain images, and the STACOM'11 myocardial tracking challenge. Accuracy measurements demonstrate that our method compares favourably with state-of-the-art methods whilst achieving volume preservation.

A Dempster-Shafer approach to trustworthy AI with application to fetal brain MRI segmentation

“…Atlas-based segmentation algorithms are anatomically-constrained due to the spatial smoothness that is imposed to the spatial transformation used to compute the segmentation. In practice, this is achieved thanks to the parameterization of the spatial transformation and the regularization loss in the registration optimization problem [6,19,39]. Therefore, if implemented correctly, atlas-based automatic segmentations can inherit from the anatomical prior represented by segmentation atlases.…”

Biomechanics-Informed Neural Networks for Myocardial Motion Tracking in MRI

“…To take into account the invertibility of the transformation, many diffeomorphic registration methods have been proposed to parameterise the displacement field as the integral of a time-varying velocity field or stationary velocity field (SVF) [4,1], which mathematically guarantees diffeomorphism. Volumetric preservation is also a characteristic that is often desired for soft tissue tracking such as myocardium tracking to ensure that the total volume of myocardium keeps constant during image registration [12,18,26]. Such incompressible registration methods either relax det(J Φ (x)) = 1 as a soft constraint [26] (where J Φ is the Jacobian matrix of the transformation Φ), or use specific parameterisation for the displacement field or SVF [12,18].…”

“…Volumetric preservation is also a characteristic that is often desired for soft tissue tracking such as myocardium tracking to ensure that the total volume of myocardium keeps constant during image registration [12,18,26]. Such incompressible registration methods either relax det(J Φ (x)) = 1 as a soft constraint [26] (where J Φ is the Jacobian matrix of the transformation Φ), or use specific parameterisation for the displacement field or SVF [12,18]. In addition, population statistics have also been investigated to incorporate prior knowledge for the registration process [15,24].…”